Ομάδα SU(3)
Ομάδα SU(3) SU(3) Group thumb|300px| [[Ομαδοθεωρία ---- Αλγεβρική Ομάδα Γενική Γραμμική Ομάδα Ορθογώνια Ομάδα Μοναδιακή Ομάδα ---- Μαθηματική Αναπαράσταση Μαθηματική Μήτρα ]] thumb|300px|[[Τριχρωμία, Χρωματικός Κύκλος Νεύτωνα]] - Μία Ομάδα. Ετυμολογία Η ονομασία "ομάδα" σχετίζεται ετυμολογικά με την λέξη '"ομού". Περιγραφή Η ομάδα SU (3) έχει οκτώ γεννήτορες, και διαπιστώθηκε ότι αυτό σημαίνει ότι υπάρχουν και οκτώ είδη γλοιονίων που προβλέπεται σωστά από τη θεωρία. The generators of '''su'(3), T'', in the defining representation, are: : T_a = \frac{\lambda_a }{2}.\, where λ the Gell-Mann matrices, are the SU(3) analog of the Pauli matrices for SU(2): : Note that they span all traceless Hermitian matrices as required. These obey the relations : \leftT_b \right = i \sum_{c=1}^8{f_{abc} T_c} \, : \{T_a, T_b\} = \frac{1}{3}\delta_{ab} + \sum_{c=1}^8{d_{abc} T_c} \, (or, the same, \{\lambda_a, \lambda_b\} = \frac{4}{3}\delta_{ab} + 2\sum_{c=1}^8{d_{abc} \lambda_c} \, ) The ''f are the structure constants, given by: : f_{123} = 1 \, : f_{147} = -f_{156} = f_{246} = f_{257} = f_{345} = -f_{367} = \frac{1}{2} \, : f_{458} = f_{678} = \frac{\sqrt{3}}{2}, \, and all other f_{abc} not related to these by permutation are zero. The d'' take the values: : d_{118} = d_{228} = d_{338} = -d_{888} = \frac{1}{\sqrt{3}} \, : d_{448} = d_{558} = d_{668} = d_{778} = -\frac{1}{2\sqrt{3}} \, : d_{146} = d_{157} = -d_{247} = d_{256} = d_{344} = d_{355} = -d_{366} = -d_{377} = \frac{1}{2}. \, Εφαρμογή The color group '''SU'(3) corresponds to the local symmetry whose gauging gives rise to QCD. The electric charge labels a representation of the local symmetry group U'(1) which is gauged to give QED: this is an abelian group. If one considers a version of QCD with ''Nf flavors of massless quarks, then there is a global (chiral) flavor symmetry group 'SU'L(Nf) × 'SU'R(Nf) × 'U'B(1) × 'U'A(1). The chiral symmetry is spontaneously broken by the QCD vacuum to the vector (L+R) 'SU'V(Nf) with the formation of a chiral condensate. The vector symmetry, 'U'B(1) corresponds to the baryon number of quarks and is an exact symmetry. The axial symmetry 'U'A(1) is exact in the classical theory, but broken in the quantum theory, an occurrence called an anomaly. Gluon field configurations called instantons are closely related to this anomaly. There are two different types of '''SU(3) symmetry: there is the symmetry that acts on the different colors of quarks, and this is an exact gauge symmetry mediated by the gluons, and there is also a flavor symmetry which rotates different flavors of quarks to each other, or flavor '''SU'(3). Flavor '''SU'(3) is an approximate symmetry of the vacuum of QCD, and is not a fundamental symmetry at all. It is an accidental consequence of the small mass of the three lightest quarks. In the QCD vacuum there are vacuum condensates of all the quarks whose mass is less than the QCD scale. This includes the up and down quarks, and to a lesser extent the strange quark, but not any of the others. The vacuum is symmetric under SU(2) isospin rotations of up and down, and to a lesser extent under rotations of up, down and strange, or full flavor group SU(3), and the observed particles make isospin and SU(3) multiplets. The approximate flavor symmetries do have associated gauge bosons, observed particles like the rho and the omega, but these particles are nothing like the gluons and they are not massless. They are emergent gauge bosons in an approximate string description of QCD. Lagrangian The dynamics of the quarks and gluons are controlled by the quantum chromodynamics Lagrangian. The gauge invariant QCD Lagrangian is :: \mathcal{L}_\mathrm{QCD} = \bar{\psi}_i\left(i (\gamma^\mu D_\mu)_{ij} - m\, \delta_{ij}\right) \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a where \psi_i(x) \, is the quark field, a dynamical function of space-time, in the fundamental representation of the [[SU(3)|'SU'(3)]] gauge group, indexed by i,\,j,\,\ldots ; G^a_\mu(x) \, are the gluon fields, also a dynamical function of space-time, in the adjoint representation of the SU(3) gauge group, indexed by a'', ''b, .... The γμ are Dirac matrices connecting the spinor representation to the vector representation of the Lorentz group. The symbol G^a_{\mu \nu} \, represents the gauge invariant gluonic field strength tensor, analogous to the electromagnetic field strength tensor, F''μ, ν, in Electrodynamics. It is given by :: G^a_{\mu \nu} = \partial_\mu G^a_{\nu} - \partial_\nu G^a_\mu - g f^{abc} G^b_\mu G^c_\nu \,, where ''fabc are the structure constants of SU(3). Note that the rules to move-up or pull-down the a'', ''b, or c'' indexes are ''trivial, (+......+), so that fabc = fabc = f'' b=''bc|p=''a'' whereas for the μ'' or ''ν indexes one has the non-trivial relativistic rules, corresponding e.g. to the signature (+---). Furthermore, for mathematicians, according to this formula the gluon color field can be represented by a SU(3)-Lie algebra-valued "curvature"-2-form \mathbf G=\mathrm d\mathbf {\tilde G}-g\,\mathbf {\tilde G}\wedge \mathbf {\tilde G}\,, where \mathbf {\tilde G} is a "vector potential"-1-form corresponding to G''' and \wedge is the (antisymmetric) "wedge product" of this algebra, producing the "structure constants" fabc. The Cartan-derivative of the field form (i.e. essentially the divergence of the field) would be zero in the absence of the "gluon terms", i.e. those ~ g, which represent the non-abelian character of the '''SU(3). The constants m'' and ''g control the quark mass and coupling constants of the theory, subject to renormalization in the full quantum theory. An important theoretical notion concerning the final term of the above Lagrangian is the Wilson loop variable. This loop variable plays a most-important role in discretized forms of the QCD (see lattice QCD), and more generally, it distinguishes confined and deconfined states of a gauge theory. It was introduced by the Nobel prize winner Kenneth G. Wilson and is treated in a separate article. Fields Quarks are massive spin-1/2 fermions which carry a color charge whose gauging is the content of QCD. Quarks are represented by Dirac fields in the fundamental representation 3''' of the gauge group [[SU(3)|'''SU(3)]]. They also carry electric charge (either -1/3 or 2/3) and participate in weak interactions as part of weak isospin doublets. They carry global quantum numbers including the baryon number, which is 1/3 for each quark, hypercharge and one of the flavor quantum numbers. Gluons are spin-1 bosons which also carry color charges, since they lie in the adjoint representation 8''' of '''SU(3). They have no electric charge, do not participate in the weak interactions, and have no flavor. They lie in the singlet representation 1 of all these symmetry groups. Every quark has its own antiquark. The charge of each antiquark is exactly the opposite of the corresponding quark. Dynamics According to the rules of quantum field theory, and the associated Feynman diagrams, the above theory gives rise to three basic interactions: a quark may emit (or absorb) a gluon, a gluon may emit (or absorb) a gluon, and two gluons may directly interact. This contrasts with QED, in which only the first kind of interaction occurs, since photons have no charge. Diagrams involving Faddeev-Popov ghosts must be considered too (except in the unitarity gauge). Υποσημειώσεις Εσωτερική Αρθρογραφία *Ομάδα *Ομαδοθεωρία *Ορθογώνια Ομάδα *Μοναδιακή Ομάδα *Ετεροτική Ομάδα *Αναπαράσταση *Μήτρα Βιβλιογραφία * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *[ ] *[ ] Category: Μαθηματικές Ομάδες